A coupled system of nonlocal fractional differential equations with coupled and uncoupled slit-strips integral boundary conditions
This paper is concerned with the existence and uniqueness of solutions for a coupled system of fractional differential equations with coupled and uncoupled slit-strips integral boundary conditions. The existence and uniqueness of solutions is established by Banach’s contraction principle, while the...
Gespeichert in:
| Datum: | 2016 |
|---|---|
| Hauptverfasser: | , |
| Format: | Artikel |
| Sprache: | English |
| Veröffentlicht: |
Інститут математики НАН України
2016
|
| Schriftenreihe: | Нелінійні коливання |
| Online Zugang: | http://dspace.nbuv.gov.ua/handle/123456789/177259 |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Zitieren: | A coupled system of nonlocal fractional differential equations with coupled and uncoupled slit-strips integral boundary conditions / B. Ahmad, S.K. Ntouyas // Нелінійні коливання. — 2016. — Т. 19, № 3. — С. 291-310 — Бібліогр.: 37 назв. — англ. |
Institution
Digital Library of Periodicals of National Academy of Sciences of Ukraine| id |
irk-123456789-177259 |
|---|---|
| record_format |
dspace |
| spelling |
irk-123456789-1772592021-02-14T01:26:23Z A coupled system of nonlocal fractional differential equations with coupled and uncoupled slit-strips integral boundary conditions Ahmad, B. Ntouyas, S.K. This paper is concerned with the existence and uniqueness of solutions for a coupled system of fractional differential equations with coupled and uncoupled slit-strips integral boundary conditions. The existence and uniqueness of solutions is established by Banach’s contraction principle, while the existence of solutions is derived by using Leray – Schauder’s alternative. The results are explained with the aid of examples. Розглядається iснування та єдинiсть розв’язкiв з’єднаних систем нелокальних диференцiальних рiвнянь дробового порядку зi з’єднаними та нез’єднаними розщепленими смугами в iнтегральних граничних умовах. Iснування та єдинiсть розв’язкiв встановлено за допомогою теореми Банаха про стискаючi вiдображення. Iснування розв’язкiв доведено з використанням альтернативи Лерея – Шаудера. Результати пояснено за допомогою прикладiв. 2016 Article A coupled system of nonlocal fractional differential equations with coupled and uncoupled slit-strips integral boundary conditions / B. Ahmad, S.K. Ntouyas // Нелінійні коливання. — 2016. — Т. 19, № 3. — С. 291-310 — Бібліогр.: 37 назв. — англ. 1562-3076 http://dspace.nbuv.gov.ua/handle/123456789/177259 517.9 en Нелінійні коливання Інститут математики НАН України |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| collection |
DSpace DC |
| language |
English |
| description |
This paper is concerned with the existence and uniqueness of solutions for a coupled system of fractional differential equations with coupled and uncoupled slit-strips integral boundary conditions. The existence and uniqueness of solutions is established by Banach’s contraction principle, while the existence of solutions is derived by using Leray – Schauder’s alternative. The results are explained with the aid of examples. |
| format |
Article |
| author |
Ahmad, B. Ntouyas, S.K. |
| spellingShingle |
Ahmad, B. Ntouyas, S.K. A coupled system of nonlocal fractional differential equations with coupled and uncoupled slit-strips integral boundary conditions Нелінійні коливання |
| author_facet |
Ahmad, B. Ntouyas, S.K. |
| author_sort |
Ahmad, B. |
| title |
A coupled system of nonlocal fractional differential equations with coupled and uncoupled slit-strips integral boundary conditions |
| title_short |
A coupled system of nonlocal fractional differential equations with coupled and uncoupled slit-strips integral boundary conditions |
| title_full |
A coupled system of nonlocal fractional differential equations with coupled and uncoupled slit-strips integral boundary conditions |
| title_fullStr |
A coupled system of nonlocal fractional differential equations with coupled and uncoupled slit-strips integral boundary conditions |
| title_full_unstemmed |
A coupled system of nonlocal fractional differential equations with coupled and uncoupled slit-strips integral boundary conditions |
| title_sort |
coupled system of nonlocal fractional differential equations with coupled and uncoupled slit-strips integral boundary conditions |
| publisher |
Інститут математики НАН України |
| publishDate |
2016 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/177259 |
| citation_txt |
A coupled system of nonlocal fractional differential equations with coupled and uncoupled slit-strips integral boundary conditions / B. Ahmad, S.K. Ntouyas // Нелінійні коливання. — 2016. — Т. 19, № 3. — С. 291-310 — Бібліогр.: 37 назв. — англ. |
| series |
Нелінійні коливання |
| work_keys_str_mv |
AT ahmadb acoupledsystemofnonlocalfractionaldifferentialequationswithcoupledanduncoupledslitstripsintegralboundaryconditions AT ntouyassk acoupledsystemofnonlocalfractionaldifferentialequationswithcoupledanduncoupledslitstripsintegralboundaryconditions AT ahmadb coupledsystemofnonlocalfractionaldifferentialequationswithcoupledanduncoupledslitstripsintegralboundaryconditions AT ntouyassk coupledsystemofnonlocalfractionaldifferentialequationswithcoupledanduncoupledslitstripsintegralboundaryconditions |
| first_indexed |
2025-07-15T15:17:55Z |
| last_indexed |
2025-07-15T15:17:55Z |
| _version_ |
1837726617931087872 |
| fulltext |
UDC 517.9
A COUPLED SYSTEM OF NONLOCAL FRACTIONAL DIFFERENTIAL
EQUATIONS WITH COUPLED AND UNCOUPLED SLIT-STRIPS
INTEGRAL BOUNDARY CONDITIONS
З’ЄДНАНI СИСТЕМИ НЕЛОКАЛЬНИХ ДИФЕРЕНЦIАЛЬНИХ
РIВНЯНЬ ДРОБОВОГО ПОРЯДКУ ЗI З’ЄДНАНИМИ
ТА НЕЗ’ЄДНАНИМИ РОЗЩЕПЛЕНИМИ СМУГАМИ
В IНТЕГРАЛЬНИХ ГРАНИЧНИХ УМОВАХ
B. Ahmad
King Abdulaziz Univ.
P. O. Box 80203, Jeddah 21589, Saudi Arabia
e-mail: bashirahmad_qau@yahoo.com
S. K. Ntouyas
Univ. Ioannina
451 10 Ioannina, Greece
and King Abdulaziz Univ.
P. O. Box 80203, Jeddah 21589, Saudi Arabia
e-mail: sntouyas@uoi.gr
This paper is concerned with the existence and uniqueness of solutions for a coupled system of fractional
differential equations with coupled and uncoupled slit-strips integral boundary conditions. The existence
and uniqueness of solutions is established by Banach’s contraction principle, while the existence of soluti-
ons is derived by using Leray – Schauder’s alternative. The results are explained with the aid of examples.
Розглядається iснування та єдинiсть розв’язкiв з’єднаних систем нелокальних диференцiаль-
них рiвнянь дробового порядку зi з’єднаними та нез’єднаними розщепленими смугами в iнте-
гральних граничних умовах. Iснування та єдинiсть розв’язкiв встановлено за допомогою тео-
реми Банаха про стискаючi вiдображення. Iснування розв’язкiв доведено з використанням аль-
тернативи Лерея – Шаудера. Результати пояснено за допомогою прикладiв.
1. Introduction. The study of boundary-value problems for linear and nonlinear differential
equations is a popular field of research and finds extensive applications in a variety of disciplines
of pure and applied sciences. The investigation of boundary-value problems of fractional-order
has recently picked up a great momentum and a variety of results of diverse interest, ranging
from theoretical to application aspects, are available in the literature on the topic. In particular,
the tools of fractional calculus have revolutionized the field of mathematical modelling and the
integer-order models in many physical and engineering phenomena have been transformed
to their fractional-order counterparts. One of the salient features accounting for this trend
is probably the nonlocal characteristic of fractional-order operators, which can describe the
hereditary properties of many important materials and processes. For examples and appli-
cations in physics, chemistry, biology, biophysics, blood flow phenomena, control theory, wave
propagation, signal and image processing, viscoelasticity, percolation, identification, fitting of
c© B. Ahmad, S. K. Ntouyas, 2016
ISSN 1562-3076. Нелiнiйнi коливання, 2016, т . 19, N◦ 3 291
292 B. AHMAD, S. K. NTOUYAS
experimental data, economics etc., we refer the reader to the books [1 – 3]. For some recent
work on the topic, see [4 – 24] and the references therein. In a recent paper [25], the authors
discussed some new fractional boundary-value problems with slit-strips conditions.
The investigation of coupled systems of fractional order differential equations is also very
significant as such systems appear in a variety of problems of applied nature, especially in biosci-
ences. For details and examples, the reader is referred to the papers [26 – 32] and the references
cited therein.
In this paper, motivated by [25], we study a coupled system of nonlinear fractional differenti-
al equations:
cDqx(t) = f(t, x(t), y(t)), t ∈ [0, 1], 1 < q ≤ 2,
cDpy(t) = g(t, x(t), y(t)), t ∈ [0, 1], 1 < p ≤ 2,
(1.1)
supplemented with coupled and uncoupled slit-strips type integral boundary conditions respecti-
vely given by
x(0) = 0, x(ζ) = a
η∫
0
y(s)ds+ b
1∫
ξ
y(s)ds, 0 < η < ζ < ξ < 1,
y(0) = 0, y(ζ) = a
η∫
0
x(s)ds+ b
1∫
ξ
x(s)ds, 0 < η < ζ < ξ < 1,
(1.2)
and
x(0) = 0, x(ζ) = a
η∫
0
x(s)ds+ b
1∫
ξ
x(s)ds, 0 < η < ζ < ξ < 1,
y(0) = 0, y(ζ) = a
η∫
0
y(s)ds+ b
1∫
ξ
y(s)ds, 0 < η < ζ < ξ < 1,
(1.3)
where cDq, cDp denote the Caputo fractional derivative of order q and p respectively, f, g :
[0, 1]× R× R → R are given continuous functions, and a, b are real constants.
Here we remark that the differential equations with integral boundary conditions consti-
tute an important class of boundary-value problems. The concept of coupled and uncoupled
integral boundary conditions introduced in this paper is new. We can interpret these conditions
physically as the contribution due to finite strips of arbitrary lengths on the given interval is
related to the value of the unknown function at an arbitrary (nonlocal) position in the region
off these strips. The applications of strip-slit boundary conditions, for instance, can be found in
the works [33 – 36].
The paper is organized as follows. In Section 2, we present the main results for a coupled
system of nonlinear fractional differential equations with coupled slit-strips integral boundary
ISSN 1562-3076. Нелiнiйнi коливання, 2016, т . 19, N◦ 3
A COUPLED SYSTEM OF NONLOCAL FRACTIONAL DIFFERENTIAL EQUATIONS . . . 293
conditions while the results for uncoupled integral boundary conditions are discussed in Secti-
on 3. Our results rely on the standard tools of the fixed point theory and are well illustrated
with the aid of examples.
2. Coupled slit-strips integral boundary conditions case. First of all, we recall definitions of
fractional integral and derivative [1, 2].
Definition 2.1. The Riemann – Liouville fractional integral of order q for a continuous functi-
on g is defined as
Iqg(t) =
1
Γ(q)
t∫
0
g(s)
(t− s)1−q
ds, q > 0,
provided the integral exists.
Definition 2.2. For at least n-times continuously differentiable function g : [0,∞) → R, the
Caputo derivative of fractional order q is defined as
cDqg(t) =
1
Γ(n− q)
t∫
0
(t− s)n−q−1g(n)(s)ds, n− 1 < q < n, n = [q] + 1,
where [q] denotes the integer part of the real number q.
Now we prove an auxiliary result which is pivotal to define the solution for the problem
(1.1), (1.2).
Lemma 2.1 (Auxiliary lemma). Given φ, ψ ∈ C([0, 1],R), the following system:
cDqx(t) = φ(t), t ∈ [0, 1], 1 < q ≤ 2,
cDpy(t) = ψ(t), t ∈ [0, 1], 1 < p ≤ 2,
(2.1)
x(0) = 0, x(ζ) = a
η∫
0
y(s)ds+ b
1∫
ξ
y(s)ds, 0 < η < ζ < ξ < 1,
y(0) = 0, y(ζ) = a
η∫
0
x(s)ds+ b
1∫
ξ
x(s)ds, 0 < η < ζ < ξ < 1,
can be written in the equivalent integral equations
x(t) =
t
ζ2 −∆2
ζ{a η∫
0
s∫
0
(s− τ)p−1
Γ(p)
ψ(τ) dτds+ b
1∫
ξ
s∫
0
(s− τ)p−1
Γ(p)
ψ(τ) dτds−
−
ζ∫
0
(ζ − s)q−1
Γ(q)
φ(s) ds
}
+ ∆
{
a
η∫
0
s∫
0
(s− τ)q−1
Γ(q)
φ(τ) dτds+
ISSN 1562-3076. Нелiнiйнi коливання, 2016, т . 19, N◦ 3
294 B. AHMAD, S. K. NTOUYAS
+b
1∫
ξ
s∫
0
(s− τ)q−1
Γ(q)
φ(τ) dτds−
ζ∫
0
(ζ − s)p−1
Γ(p)
ψ(s) ds
}+
+
t∫
0
(t− s)q−1
Γ(q)
φ(s)ds, (2.2)
y(t) =
t
ζ2 −∆2
∆
{
a
η∫
0
s∫
0
(s− τ)p−1
Γ(p)
ψ(τ) dτds+ b
1∫
ξ
∫ s
0
(s− τ)p−1
Γ(p)
ψ(τ) dτds−
−
ζ∫
0
(ζ − s)q−1
Γ(q)
φ(s) ds
}
+ ζ
{
a
η∫
0
s∫
0
(s− τ)q−1
Γ(q)
φ(τ) dτds+
+b
1∫
ξ
s∫
0
(s− τ)q−1
Γ(q)
φ(τ) dτds−
ζ∫
0
(ζ − s)p−1
Γ(p)
ψ(s)ds
}+
+
t∫
0
(t− s)p−1
Γ(p)
ψ(s) ds, (2.3)
where
∆ =
[
aη2 + b
(
1− ξ2
)]
/2 6= 0. (2.4)
Proof. It is well known that the general solution of the fractional differential equations in
(2.1) can be written as
x(t) = c0 + c1t+
t∫
0
(t− s)q−1
Γ(q)
φ(s) ds, (2.5)
y(t) = c2 + c3t+
t∫
0
(t− s)p−1
Γ(p)
ψ(s) ds, (2.6)
where c0, c1 ∈ R are arbitrary constants.
Applying the conditions x(0) = 0, y(0) = 0, it is found that c0 = 0, c2 = 0. In view of the
nonlocal conditions
x(ζ) = a
η∫
0
y(s)ds+ b
1∫
ξ
y(s)ds, y(ζ) = a
η∫
0
x(s)ds+ b
1∫
ξ
x(s)ds,
ISSN 1562-3076. Нелiнiйнi коливання, 2016, т . 19, N◦ 3
A COUPLED SYSTEM OF NONLOCAL FRACTIONAL DIFFERENTIAL EQUATIONS . . . 295
we obtain a system of equations
ζc1 −∆c3 = a
η∫
0
s∫
0
(s− τ)p−1
Γ(p)
ψ(τ) dτds+ b
1∫
ξ
s∫
0
(s− τ)p−1
Γ(p)
ψ(τ) dτds−
−
ζ∫
0
(ζ − s)q−1
Γ(q)
φ(s)ds,
−∆c1 + ζc3 = a
η∫
0
s∫
0
(s− τ)q−1
Γ(q)
φ(τ) dτds+ b
1∫
ξ
s∫
0
(s− τ)q−1
Γ(q)
φ(τ) dτds−
−
ζ∫
0
(ζ − s)p−1
Γ(p)
ψ(s) ds,
where
∆ =
[
aη2 + b
(
1− ξ2
)]
/2.
Solving the system (2.7), (2.8), we have
c1 =
1
ζ2 −∆2
ζ{a η∫
0
s∫
0
(s− τ)p−1
Γ(p)
ψ(τ) dτds+ b
1∫
ξ
s∫
0
(s− τ)p−1
Γ(p)
ψ(τ) dτds−
−
ζ∫
0
(ζ − s)q−1
Γ(q)
φ(s)ds
}
+ ∆
{
a
η∫
0
s∫
0
(s− τ)q−1
Γ(q)
φ(τ) dτds+
+b
1∫
ξ
s∫
0
(s− τ)q−1
Γ(q)
φ(τ) dτds−
ζ∫
0
(ζ − s)p−1
Γ(p)
ψ(s) ds
} ,
c3 =
1
ζ2 −∆2
∆
{
a
η∫
0
s∫
0
(s− τ)p−1
Γ(p)
ψ(τ) dτds+ b
1∫
ξ
s∫
0
(s− τ)p−1
Γ(p)
ψ(τ) dτds−
−
ζ∫
0
(ζ − s)q−1
Γ(q)
φ(s)ds
}
+ ζ
{
a
η∫
0
s∫
0
(s− τ)q−1
Γ(q)
φ(τ) dτds+
+b
1∫
ξ
s∫
0
(s− τ)q−1
Γ(q)
φ(τ)dτ ds−
ζ∫
0
(ζ − s)p−1
Γ(p)
ψ(s) ds
} .
ISSN 1562-3076. Нелiнiйнi коливання, 2016, т . 19, N◦ 3
296 B. AHMAD, S. K. NTOUYAS
Substituting the values of c0, c1, c2, c3 in (2.5) and (2.6), we get (2.2) and (2.3). The converse
follows by direct computation.
Lemma 2.1 is proved.
2.1. Existence results. Let us introduce the space X = {x(t)|x(t) ∈ C([0, 1])} endowed
with the norm ‖x‖ = max{|x(t)|, t ∈ [0, 1]}. Obviously (X, ‖ · ‖) is a Banach space. Also
let Y = {y(t)|y(t) ∈ C([0, 1])} be endowed with the norm ‖y‖ = max{|y(t)|, t ∈ [0, 1]}.
Obviously the product space (X×Y, ‖(x, y)‖) is a Banach space with norm ‖(x, y)‖ = ‖x‖+‖y‖.
In view of Lemma 2.1, we define an operator T : X × Y → X × Y by
T (x, y)(t) =
T1(x, y)(t)
T2(x, y)(t)
,
where
T1(x, y)(t) =
t
ζ2 −∆2
ζ{a η∫
0
s∫
0
(s− τ)p−1
Γ(p)
g(τ, x(τ), y(τ)) dτds+
+ b
1∫
ξ
s∫
0
(s− τ)p−1
Γ(p)
g(τ, x(τ), y(τ)) dτds−
ζ∫
0
(ζ − s)q−1
Γ(q)
f(s, x(s), y(s)) ds
}
+
+ ∆
{
a
η∫
0
s∫
0
(s− τ)q−1
Γ(q)
f(τ, x(τ), y(τ)) dτds+
+ b
1∫
ξ
s∫
0
(s− τ)q−1
Γ(q)
f(τ, x(τ), y(τ)) dτds−
−
ζ∫
0
(ζ − s)p−1
Γ(p)
g(s, x(s), y(s)) ds
}+
t∫
0
(t− s)q−1
Γ(q)
f(s, x(s), y(s)) ds,
T2(x, y)(t) =
t
ζ2 −∆2
∆
{
a
η∫
0
s∫
0
(s− τ)p−1
Γ(p)
g(τ, x(τ), y(τ)) dτds+
+ b
1∫
ξ
s∫
0
(s− τ)p−1
Γ(p)
g(τ, x(τ), y(τ)) dτds−
ζ∫
0
(ζ − s)q−1
Γ(q)
f(s, x(s), y(s))ds
}
+
+ ζ
{
a
η∫
0
s∫
0
(s− τ)q−1
Γ(q)
f(τ, x(τ), y(τ)) dτds+
ISSN 1562-3076. Нелiнiйнi коливання, 2016, т . 19, N◦ 3
A COUPLED SYSTEM OF NONLOCAL FRACTIONAL DIFFERENTIAL EQUATIONS . . . 297
+ b
1∫
ξ
s∫
0
(s− τ)q−1
Γ(q)
f(τ, x(τ), y(τ)) dτds−
−
ζ∫
0
(ζ − s)p−1
Γ(p)
g(s, x(s), y(s))ds
}+
t∫
0
(t− s)p−1
Γ(p)
g(s, x(s), y(s)) ds.
For the sake of convenience, we set
M1 =
1
Γ(p+ 1)
+
1
|ζ2 −∆2|
[
ζq+1
Γ(q + 1)
+ |∆||a| ηq+1
Γ(q + 2)
+ |∆||b| 1− ξ
q+1
Γ(q + 2)
]
, (2.7)
M2 =
1
|ζ2 −∆2|
[
ζ|a| ηp+1
Γ(p+ 2)
+ ζ|b| 1− ξ
p+1
Γ(p+ 2)
+ |∆| ζp
Γ(p+ 1)
]
, (2.8)
M3 =
1
|ζ2 −∆2|
[
ζ|a| ηq+1
Γ(q + 2)
+ ζ|b| 1− ξ
q+1
Γ(q + 2)
+ |∆| ζq
Γ(q + 1)
]
, (2.9)
M4 =
1
Γ(p+ 1)
+
1
|ζ2 −∆2|
[
ζp+1
Γ(p+ 1)
+ |∆||a| ηp+1
Γ(p+ 2)
+ |∆||b| 1− ξ
p+1
Γ(p+ 2)
]
, (2.10)
and
M0 = min
{
1− (M1 +M3)k1 − (M2 +M4)λ1, 1− (M1 +M3)k2 − (M2 +M4)λ2
}
,
(2.11)
ki, λi ≥ 0, i = 1, 2.
The first result is concerned with the existence and uniqueness of solutions for the problem
(1.1), (1.2) and is based on Banach’s contraction mapping principle.
Theorem 2.1. Assume that f, g : [0, 1] × R2 → R are continuous functions and there exist
constants mi, ni, i = 1, 2, such that for all t ∈ [0, 1] and ui, vi ∈ R, i = 1, 2,
|f(t, u1, u2)− f(t, v1, v2)| ≤ m1|u1 − v1|+m2|u2 − v2|
and
|g(t, u1, u2)− g(t, v1, v2)| ≤ n1|u1 − v1|+ n2|u2 − v2|.
In addition, assume that
(M1 +M3)(m1 +m2) + (M2 +M4)(n1 + n2) < 1,
where Mi, i = 1, 2, 3, 4, are given by (2.7) – (2.10). Then the boundary-value problem (1.1), (1.2)
has a unique solution.
Proof. Define supt∈[0,1] f(t, 0, 0) = N1 < ∞ and supt∈[0,1] g(t, 0, 0) = N2 < ∞ such that
r ≥ (M1 +M3)N1 + (M2 +M4)N2
1− [(M1 +M3)(m1 +m2) + (M2 +M4)(n1 + n2)]
.
ISSN 1562-3076. Нелiнiйнi коливання, 2016, т . 19, N◦ 3
298 B. AHMAD, S. K. NTOUYAS
We show that TBr ⊂ Br, where Br = {(x, y) ∈ X × Y : ‖(x, y)‖ ≤ r}.
For (x, y) ∈ Br, we have
|T1(x, y)(t)| = max
t∈[0,1]
t
|ζ2 −∆2|
ζ
|a|
η∫
0
s∫
0
(s− τ)p−1
Γ(p)
(|g(τ, x(τ), y(τ))− g(τ, 0, 0)|+
+ |g(τ, 0, 0)|)dτds+ |b|
1∫
ξ
s∫
0
(s− τ)p−1
Γ(p)
(|g(τ, x(τ), y(τ))− g(τ, 0, 0)|+
+|g(τ, 0, 0)|)dτds+
ζ∫
0
(ζ − s)q−1
Γ(q)
(|f(s, x(s), y(s))− f(s, 0, 0)|+ |f(s, 0, 0)|)ds
+
+ |∆|
|a|
η∫
0
s∫
0
(s− τ)q−1
Γ(q)
(|f(τ, x(τ), y(τ))− f(τ, 0, 0)|+ |f(τ, 0, 0)|)dτds+
+ |b|
1∫
ξ
s∫
0
(s− τ)q−1
Γ(q)
t(|f(τ, x(τ), y(τ))− f(τ, 0, 0)|+ |f(τ, 0, 0)|) dτds+
+
ζ∫
0
(ζ − s)p−1
Γ(p)
(|g(s, x(s), y(s))− g(s, 0, 0)|+ |g(s, 0, 0)|) ds
+
+
t∫
0
(t− s)q−1
Γ(q)
(|f(s, x(s), y(s))− f(s, 0, 0)|+ |f(s, 0, 0)|)ds
≤
≤ 1
|ζ2 −∆2|
ζ
|a|
η∫
0
s∫
0
(s− τ)p−1
Γ(p)
(n1‖x‖+ n2‖y‖+N2) dτds+
+ |b|
1∫
ξ
s∫
0
(s− τ)p−1
Γ(p)
(n1‖x‖+ n2‖y‖+N2)dτds+
+
ζ∫
0
(ζ − s)q−1
Γ(q)
(m1‖x‖+m2‖y‖+N1)ds
+
+ |∆|
|a|
η∫
0
s∫
0
(s− τ)q−1
Γ(q)
(m1‖x‖+m2‖y‖+N1) dτds+
ISSN 1562-3076. Нелiнiйнi коливання, 2016, т . 19, N◦ 3
A COUPLED SYSTEM OF NONLOCAL FRACTIONAL DIFFERENTIAL EQUATIONS . . . 299
+ |b|
1∫
ξ
s∫
0
(s− τ)q−1
Γ(q)
(m1‖x‖+m2‖y‖+N1) dτds+
+
ζ∫
0
(ζ − s)p−1
Γ(p)
(n1‖x‖+ n2‖y‖+N2)ds
+
+
t∫
0
(t− s)q−1
Γ(q)
(m1‖x‖+m2‖y‖+N1) ds ≤
≤ 1
|ζ2 −∆2|
[
ζ|a| ηp+1
Γ(p+ 2)
+ ζ|b|1− ξ
p+1
Γ(p+ 2)
+ |∆| ζp
Γ(p+ 1)
]
(n1‖x‖+ n2‖y‖+N2)+
+
{
1
|ζ2 −∆2|
[
ζq+1
Γ(q + 1)
+ |∆||a| ηq+1
Γ(q + 2)
+ |∆||b| 1− ξ
q+1
Γ(q + 2)
]
+
1
Γ(q + 1)
}
×
× (m1‖x‖+m2‖y‖+N1) = M2(n1‖x‖+ n2‖y‖+N2) +M1(m1‖x‖+m2‖y‖+N1) =
= (M2n1 +M1m1)‖x‖+ (M2n2 +M1m2)‖y‖+M2N2 +M1N1 ≤
≤ (M2n1 +M1m1 +M2n2 +M1m2)r +M2N2 +M1N1.
In the same way, we can obtain that
|T2(x, y)(t)| ≤
{
1
|ζ2 −∆2|
[
|∆||a| ηp+1
Γ(p+ 2)
+ |∆||b| 1− ξ
p+1
Γ(p+ 2)
+ ζ
ζp+1
Γ(p+ 1)
]
+
1
Γ(q + 1)
}
×
× (n1‖x‖+ n2‖y‖+N2) +
1
|ζ2 −∆2|
×
×
[
|∆| ζq
Γ(q + 1)
+ ζ|a| ηq+1
Γ(q + 2)
+ ζ|b| 1− ξ
q+1
Γ(q + 2)
]
(m1‖x‖+m2‖y‖+N1) =
= M4(n1‖x‖+ n2‖y‖+N2) +M3(m1‖x‖+m2‖y‖+N1) =
= (M4n1 +M3m1)‖x‖+ (M4n2 +M3m2)‖y‖+M4N2 +M3N1 ≤
≤ (M4n1 +M3m1 +M4n2 +M3m2)r +M4N2 +M3N1.
Consequently, ‖T (x, y)(t)‖ ≤ r.
Now for (x2, y2), (x1, y1) ∈ X × Y, and for any t ∈ [0, 1], we get
|T1(x2, y2)(t)− T1(x1, y1)(t)| ≤
≤ t
|ζ2 −∆2|
ζ
|a|
η∫
0
s∫
0
(s− τ)p−1
Γ(p)
|g(τ, x2(τ), y2(τ))− g(τ, x1(τ), y1(τ))| dτds+
ISSN 1562-3076. Нелiнiйнi коливання, 2016, т . 19, N◦ 3
300 B. AHMAD, S. K. NTOUYAS
+ |b|
1∫
ξ
s∫
0
(s− τ)p−1
Γ(p)
|g(τ, x2(τ), y2(τ))− g(τ, x1(τ), y1(τ))|dτds+
+
ζ∫
0
(ζ − s)q−1
Γ(q)
|f(s, x2(s), y2(s))− f(s, x1(s), y1(s))|ds
+
+ |∆|
|a|
η∫
0
s∫
0
(s− τ)q−1
Γ(q)
|f(τ, x2(τ), y2(τ))− f(τ, x1(τ), y1(τ))|dτds+
+ |b|
1∫
ξ
s∫
0
(s− τ)q−1
Γ(q)
|f(τ, x2(τ), y2(τ))− f(τ, x1(τ), y1(τ))| dτds+
+
ζ∫
0
(ζ − s)p−1
Γ(p)
|g(s, x2(s), y2(s))− g(s, x1(s), y1(s))|ds
+
+
t∫
0
(t− s)q−1
Γ(q)
|g(s, x2(s), y2(s))− g(s, x1(s), y1(s))| ds ≤
1
|ζ2 −∆2|
×
×
[
ζ|a| ηp+1
Γ(p+ 2)
+ ζ|b| 1− ξ
p+1
Γ(p+ 2)
+ |∆| ζp
Γ(p+ 1)
]
(n1‖x2 − x1‖+ n2‖y2 − y1‖)+
+
{
1
|ζ2 −∆2|
[
ζq+1
Γ(q + 1)
+ |∆||a| ηq+1
Γ(q + 2)
+ |∆||b| 1− ξ
q+1
Γ(q + 2)
]
+
1
Γ(q + 1)
}
×
× (m1‖x2 − x1‖+m2‖y2 − y1‖) ≤ M2(n1‖x2 − x1‖+ n2‖y2 − y1‖)+
+M1(m1‖x2 − x1‖+m2‖y2 − y1‖) = (M2n1 +M1m1)‖x2 − x1‖+
+ (M2n2 +M1m2)‖y2 − y1‖,
and consequently we obtain
‖T1(x2, y2)(t)− T1(x1, y1)‖ ≤ (M2n1+M1m1+M2n2+M1m2)[‖x2 − x1‖+ ‖y2 − y1‖]. (2.12)
Similarly,
‖T2(x2, y2)(t)−T2(x1, y1)‖ ≤ (M4n1+M3m1+M4n2+M3m2)[‖x2 − x1‖+ ‖y2 − y1‖]. (2.13)
It follows from (2.12) and (2.13) that
‖T (x2, y2)(t)−T (x1, y1)(t)‖ ≤ [(M1+M3)(m1+m2)+(M2+M4)(n1+n2)](‖u2−u1‖+‖v2−v1‖).
ISSN 1562-3076. Нелiнiйнi коливання, 2016, т . 19, N◦ 3
A COUPLED SYSTEM OF NONLOCAL FRACTIONAL DIFFERENTIAL EQUATIONS . . . 301
Since (M1 +M3)(m1 +m2) + (M2 +M4)(n1 + n2) < 1, therefore, T is a contraction operator.
So, by Banach’s fixed point theorem, the operator T has a unique fixed point, which is the
unique solution of problem (1.1), (1.2).
Theorem 2.1 is proved.
In the next result, we prove the existence of solutions for the problem (1.1), (1.2) by applying
Leray – Schauder alternative.
Lemma 2.2 (Leray – Schauder alternative, [37, p. 4]). Let F : E → E be a completely conti-
nuous operator (i.e., a map that restricted to any bounded set in E is compact). Let
E(F ) = {x ∈ E : x = λF (x) for some 0 < λ < 1}.
Then either the set E(F ) is unbounded, or F has at least one fixed point.
Theorem 2.2. Assume that there exist real constants ki, λi ≥ 0, i = 1, 2, and k0 > 0, λ0 > 0
such that for any xi ∈ R, i = 1, 2, we have
|f(t, x1, x2)| ≤ k0 + k1|x1|+ k2|x2|,
|g(t, x1, x2)| ≤ λ0 + λ1|x1|+ λ2|x2|.
In addition it is assumed that
(M1 +M3)k1 + (M2 +M4)λ1 < 1 and (M1 +M3)k2 + (M2 +M4)λ2] < 1,
where Mi, i = 1, 2, 3, 4, are given by (2.7) – (2.10). Then there exists at least one solution for the
boundary-value problem (1.1), (1.2).
Proof. First, we show that the operator T : X × Y → X × Y is completely continuous.
By continuity of functions f and g, the operator T is continuous.
Let Ω ⊂ X × Y be bounded. Then there exist positive constants L1 and L2 such that
|f(t, x(t), y(t))| ≤ L1, |g(t, x(t), y(t))| ≤ L2 ∀(x, y) ∈ Ω.
Then for any (x, y) ∈ Ω we have
|T1(x, y)(t)| ≤ t
ζ2 −∆2
ζ{|a| η∫
0
s∫
0
(s− τ)p−1
Γ(p)
|g(τ, x(τ), y(τ))|dτds+
+ |b|
1∫
ξ
s∫
0
(s− τ)p−1
Γ(p)
|g(τ, x(τ), y(τ))|dτds+
+
ζ∫
0
(ζ − s)q−1
Γ(q)
|f(s, x(s), y(s))|ds
}
+
+ |∆|
{
|a|
η∫
0
s∫
0
(s− τ)q−1
Γ(q)
|f(τ, x(τ), y(τ))| dτds+
ISSN 1562-3076. Нелiнiйнi коливання, 2016, т . 19, N◦ 3
302 B. AHMAD, S. K. NTOUYAS
+|b|
1∫
ξ
s∫
0
(s− τ)q−1
Γ(q)
|f(τ, x(τ), y(τ))|dτds+
ζ∫
0
(ζ − s)p−1
Γ(p)
|g(s, x(s), y(s))|ds
}+
+
t∫
0
(t− s)q−1
Γ(q)
|f(s, x(s), y(s))|ds ≤
≤ 1
|ζ2 −∆2|
[
ζ|a| ηp+1
Γ(p+ 2)
+ ζ|b| 1− ξ
p+1
Γ(p+ 2)
+ |∆| ζp
Γ(p+ 1)
]
L2+
+
{
1
|ζ2 −∆2|
[
ζq+1
Γ(q + 1)
+ |∆||a| ηq+1
Γ(q + 2)
+ |∆||b| 1− ξ
q+1
Γ(q + 2)
]
+
1
Γ(q + 1)
}
L1,
which implies that
‖T1(x, y)‖ ≤ 1
|ζ2 −∆2|
[
ζ|a| ηp+1
Γ(p+ 2)
+ ζ|b| 1− ξ
p+1
Γ(p+ 2)
+ |∆| ζp
Γ(p+ 1)
]
L2+
+
{
1
|ζ2 −∆2|
[
ζq+1
Γ(q + 1)
+ |∆||a| ηq+1
Γ(q + 2)
+ |∆||b| 1− ξ
q+1
Γ(q + 2)
]
+
1
Γ(q + 1)
}
L1 =
= M2L2 +M1L1.
Similarly, we get
‖T2(x, y)‖ ≤
{
1
|ζ2 −∆2|
[
|∆||a| ηp+1
Γ(p+ 2)
+ |∆||b| 1− ξ
p+1
Γ(p+ 2)
+ ζ
ζp+1
Γ(p+ 1)
]
+
1
Γ(q + 1)
}
L2+
+
1
|ζ2 −∆2|
[
|∆| ζq
Γ(q + 1)
+ ζ|a| ηq+1
Γ(q + 2)
+ ζ|b| 1− ξ
q+1
Γ(q + 2)
]
L1 =M4L2 +M3L1.
Thus, it follows from the above inequalities that the operator T is uniformly bounded.
Next, we show that T is equicontinuous. Let t1, t2 ∈ [0, 1] with t1 < t2. Then we have
|T1(x(t2), y(t2))− T1(x(t1), y(t1))| ≤
≤ 1
Γ(q)
t1∫
0
[(t2 − s)q−1 − (t1 − s)q−1]|f(s, x(s), y(s))|ds+
+
1
Γ(q)
t2∫
t1
(t2 − s)q−1|f(s, x(s), y(s))|ds+
+
t2 − t1
ζ2 −∆2
ζ{|a| η∫
0
s∫
0
(s− τ)p−1
Γ(p)
|g(τ, x(τ), y(τ))|dτds+
ISSN 1562-3076. Нелiнiйнi коливання, 2016, т . 19, N◦ 3
A COUPLED SYSTEM OF NONLOCAL FRACTIONAL DIFFERENTIAL EQUATIONS . . . 303
+ |b|
1∫
ξ
s∫
0
(s− τ)p−1
Γ(p)
|g(τ, x(τ), y(τ))| dτds+
ζ∫
0
(ζ − s)q−1
Γ(q)
|f(s, x(s), y(s))|ds
}
+
+ |∆|
{
|a|
η∫
0
s∫
0
(s− τ)q−1
Γ(q)
|f(τ, x(τ), y(τ))|dτds+
+|b|
1∫
ξ
s∫
0
(s− τ)q−1
Γ(q)
|f(τ, x(τ), y(τ))|dτds+
ζ∫
0
(ζ − s)p−1
Γ(p)
|g(s, x(s), y(s))|ds
} ≤
≤ L1
Γ(q)
t1∫
0
[
(t2 − s)q−1 − (t1 − s)q−1
]
ds+
L1
Γ(q)
t2∫
t1
(t2 − s)q−1ds+
+
t2 − t1
|ζ2 −∆2|
{[
ζ|a| ηp+1
Γ(p+ 2)
+ ζ|b| 1− ξ
p+1
Γ(p+ 2)
+ |∆| ζp
Γ(p+ 1)
]
L2+
+
[
ζq+1
Γ(q + 1)
+ |∆||a| ηq+1
Γ(q + 2)
+ |∆||b| 1− ξ
q+1
Γ(q + 2)
]
L1
}
.
Analogously, we can obtain
|T2(x(t2), y(t2))− T2(x(t1), y(t1))| ≤
≤ L2
Γ(p)
t1∫
0
[
(t2 − s)p−1 − (t1 − s)p−1
]
ds+
L2
Γ(p)
t2∫
t1
(t2 − s)p−1ds+
+
t2 − t1
|ζ2 −∆2|
{[
|∆||a| ηp+1
Γ(p+ 2)
+ |∆||b| 1− ξ
p+1
Γ(p+ 2)
+ ζ
ζp+1
Γ(p+ 1)
]
L2+
+
[
|∆| ζq
Γ(q + 1)
+ ζ|a| ηq+1
Γ(q + 2)
+ ζ|b| 1− ξ
q+1
Γ(q + 2)
]
L1
}
.
Therefore, the operator T (x, y) is equicontinuous, and thus the operator T (x, y) is completely
continuous.
Finally, it will be verified that the set E = {(x, y) ∈ X × Y |(x, y) = λT (x, y), 0 ≤ λ ≤ 1} is
bounded. Let (x, y) ∈ E , then (x, y) = λT (x, y). For any t ∈ [0, 1] we have
x(t) = λT1(x, y)(t), y(t) = λT2(x, y)(t).
Then
|x(t)| ≤ 1
ζ2 −∆2
ζ{|a| η∫
0
s∫
0
(s− τ)p−1
Γ(p)
|g(τ, x(τ), y(τ))|dτds+
ISSN 1562-3076. Нелiнiйнi коливання, 2016, т . 19, N◦ 3
304 B. AHMAD, S. K. NTOUYAS
+ |b|
1∫
ξ
s∫
0
(s− τ)p−1
Γ(p)
|g(τ, x(τ), y(τ))|dτds+
ζ∫
0
(ζ − s)q−1
Γ(q)
|f(s, x(s), y(s))|ds
}
+
+ |∆|
{
|a|
η∫
0
s∫
0
(s− τ)q−1
Γ(q)
|f(τ, x(τ), y(τ))|dτds+
+|b|
1∫
ξ
s∫
0
(s− τ)q−1
Γ(q)
|f(τ, x(τ), y(τ))|dτds+
ζ∫
0
(ζ − s)p−1
Γ(p)
|g(s, x(s), y(s))|ds
}+
+
t∫
0
(t− s)q−1
Γ(q)
|f(s, x(s), y(s))|ds ≤ 1
|ζ2 −∆2|
×
×
[
ζ|a| ηp+1
Γ(p+ 2)
+ ζ|b| 1− ξ
p+1
Γ(p+ 2)
+ |∆| ζp
Γ(p+ 1)
]
(λ0 + λ1‖x‖+ λ2‖y‖)+
+
{
1
|ζ2 −∆2|
[
ζq+1
Γ(q + 1)
+ |∆||a| ηq+1
Γ(q + 2)
+ |∆||b| 1− ξ
q+1
Γ(q + 2)
]
+
1
Γ(q + 1)
}
×
× (k0 + k1‖x‖+ k2‖y‖)
and
|y(t)| ≤
{
1
|ζ2 −∆2|
[
|∆||a| ηp+1
Γ(p+ 2)
+ |∆||b| 1− ξ
p+1
Γ(p+ 2)
+ ζ
ζp+1
Γ(p+ 1)
]
+
1
Γ(q + 1)
}
×
× (λ0 + λ1‖x‖+ λ2‖y‖) +
1
|ζ2 −∆2|
[
|∆| ζq
Γ(q + 1)
+ ζ|a| ηq+1
Γ(q + 2)
+ ζ|b| 1− ξ
q+1
Γ(q + 2)
]
×
× (k0 + k1‖x‖+ k2‖y‖).
Hence we have
‖x‖ ≤ M1(k0 + k1‖x‖+ k2‖y‖) +M2(λ0 + λ1‖x‖+ λ2‖y‖)
and
‖y‖ ≤ M3(k0 + k1‖x‖+ k2‖y‖) +M4(λ0 + λ1‖x‖+ λ2‖y‖),
which imply that
‖x‖+ ‖y‖ ≤ (M1 +M3)k0 + (M2 +M4)λ0 + [(M1 +M3)k1 + (M2 +M4)λ1]‖x‖+
+ [(M1 +M3)k2 + (M2 +M4)λ2]‖y‖.
ISSN 1562-3076. Нелiнiйнi коливання, 2016, т . 19, N◦ 3
A COUPLED SYSTEM OF NONLOCAL FRACTIONAL DIFFERENTIAL EQUATIONS . . . 305
Consequently,
‖(x, y)‖ ≤ (M1 +M3)k0 + (M2 +M4)λ0
M0
,
for any t ∈ [0, 1], where M0 is defined by (2.11), which proves that E is bounded. Thus, by Lem-
ma 2.2, the operator T has at least one fixed point. Hence the boundary-value problem (1.1),
(1.2) has at least one solution.
Theorem 2.2 is proved.
2.2. Examples. Example 2.1. Consider the following system of coupled fractional differential
equations with slit-strips integral boundary conditions:
cD5/4x(t) =
2
55
x(t) +
3
61
|y(t)|
(1 + |y(t)|)
+
3
2
, t ∈ [0, 1],
cD3/2y(t) =
1
27
| cosx(t)|
(1 + | cosx(t)|)
+
2
41
sin y(t) + 3, t ∈ [0, 1],
(2.14)
x(0) = 0, x(1/2) =
1/3∫
0
y(s)ds+
1∫
2/3
y(s)ds,
y(0) = 0, y(1/2) =
1/3∫
0
x(s)ds+
1∫
2/3
x(s)ds.
Here q = 5/4, p = 3/2, a = 1, b = 1, ζ = 1/2, η = 1/3, ξ = 2/3. With the given values,
it is found that ∆ = 1/3, m1 = 2/55, m2 = 3/61, n1 = 1/27, n2 = 2/41, M1 ' 2.731029,
M2 ' 1.397944, M3 ' 1.854888, M4 ' 2.216142, and
(M1 +M3)(m1 +m2) + (M2 +M4)(n1 + n2) ' 0.702454 < 1.
Thus all the conditions of Theorem 2.1 are satisfied. Therefore, by the conclusion of Theorem 2.1,
the problem (2.14) has a unique solution on [0, 1].
Example 2.2. Let us consider the problem (2.14) with the following values:
f(t, x(t), y(t)) =
1
2
+
2
41
sinx(t) +
2
43π
y(t) tan−1 x(t),
g(t, x(t), y(t)) =
2
3
+
1
11
x(t) +
1
17
sin y(t).
Clearly |f(t, x, y)| ≤ k0 + k1|x| + k2|y|, |g(t, x, y)| = λ0 + λ1|x| + λ2|y|, where k0 = 1/2,
k1 = 2/41, k2 = 1/43, λ0 = 2/3, λ1 = 1/11, λ2 = 1/17. Furthermore,
(M1 +M3)k1 +(M2 +M4)λ1 ' 0.552257 < 1, (M1 +M3)k2 +(M2 +M4)λ2 ' 0.319243 < 1.
Thus all the conditions for Theorem 2.2 hold true and consequently the conclusion of Theo-
rem 2.2 applies to the problem (2.14) with the given values of f(t, x, y) and g(t, x, y).
ISSN 1562-3076. Нелiнiйнi коливання, 2016, т . 19, N◦ 3
306 B. AHMAD, S. K. NTOUYAS
3. Uncoupled slit-strips integral boundary conditions case. In relation to the problem (1.1) –
(1.3), we consider the following lemma.
Lemma 3.1 (Auxiliary lemma). For χ ∈ C([0, 1],R), the unique solution of the problem
cDqx(t) = χ(t), 1 < q ≤ 2, t ∈ [0, 1],
x(0) = 0, x(ζ) = a
η∫
0
x(s)ds+ b
1∫
ξ
x(s)ds, 0 < η < ζ < ξ < 1,
(3.1)
is given by
x(t) =
t∫
0
(t− s)q−1
Γ(q)
χ(s)ds+
t
A
{
a
η∫
0
s∫
0
(s− τ)q−1
Γ(q)
χ(τ) dτds+
+ b
1∫
ξ
s∫
0
(s− τ)q−1
Γ(q)
χ(τ) dτds−
ζ∫
0
(ζ − s)q−1
Γ(q)
χ(s) d ds
}
, (3.2)
where
A = ζ − aη2
2
− b(1− ξ2)
2
6= 0. (3.3)
Proof. We just provide the outline of the proof. The general solution of the fractional di-
fferential equation in (3.1) can be written as
x(t) = e0 + e1t+
t∫
0
(t− s)q−1
Γ(q)
y(s) ds, (3.4)
where e0, e1 ∈ R are arbitrary constants. Applying the given boundary conditions, we find that
e0 = 0, and
e1 =
1
A
{
a
η∫
0
s∫
0
(s− τ)q−1
Γ(q)
y(τ) dτ ds+ b
1∫
ξ
s∫
0
(s− τ)q−1
Γ(q)
y(τ)dτds−
ζ∫
0
(ζ − s)q−1
Γ(q)
y(s) d ds
}
.
Substituting the values of e0, e1 in (3.4), we get (3.2).
Lemma 3.1 is proved.
3.1. Existence results for uncoupled case. In view of Lemma 3.1, we define an operator
T : X × Y → X × Y by
T(u, v)(t) =
T1(u, v)(t)
T2(u, v)(t)
,
ISSN 1562-3076. Нелiнiйнi коливання, 2016, т . 19, N◦ 3
A COUPLED SYSTEM OF NONLOCAL FRACTIONAL DIFFERENTIAL EQUATIONS . . . 307
where
T1(u, v)(t) =
t∫
0
(t− s)q−1
Γ(q)
f(s, u(s), v(s)) ds+
t
A
{
a
η∫
0
s∫
0
(s− τ)q−1
Γ(q)
f(τ, u(τ), v(τ)) dτds+
+ b
1∫
ξ
s∫
0
(s− τ)q−1
Γ(q)
f(τ, u(τ), v(τ)) dτds−
ζ∫
0
(ζ − s)q−1
Γ(q)
f(s, u(s), v(s)) ds
}
and
T2(u, v)(t) =
t∫
0
(t− s)p−1
Γ(p)
h(s, u(s), v(s)) ds+
t
A
{
a
η∫
0
s∫
0
(s− τ)p−1
Γ(p)
h(τ, u(τ), v(τ)) dτds+
+ b
1∫
ξ
s∫
0
(s− τ)p−1
Γ(p)
h(τ, u(τ), v(τ)) dτds−
ζ∫
0
(ζ − s)p−1
Γ(p)
f(s, u(s), v(s))ds
}
.
In the sequel, we set
µ1 =
1
Γ(q + 1)
+
1
|A|
{
|a| ηq+1
Γ(q + 2)
+ |b| 1− ξ
q+1
Γ(q + 2)
+
ζq
Γ(q + 1)
}
, (3.5)
µ2 =
1
Γ(p+ 1)
+
1
|A|
{
|a| ηp+1
Γ(p+ 2)
+ |b| 1− ξ
p+1
Γ(p+ 2)
+
ζp
Γ(p+ 1)
}
. (3.6)
Now we present the existence and uniqueness result for the problem (1.1) – (1.3). We do not
provide the proof of this result as it is similar to the one for Theorem 2.1.
Theorem 3.1. Assume that f, g : [0, 1] × R2 → R are continuous functions and there exist
constants m̄i, n̄i, i = 1, 2, such that for all t ∈ [0, 1] and ui, vi ∈ R, i = 1, 2,
|f(t, u1, u2)− g(t, v1, v2)| ≤ m̄1|u1 − v1|+ m̄2|u2 − v2|
and
|g(t, u1, u2)− h(t, v1, v2)| ≤ n̄1|u1 − v1|+ n̄2|u2 − v2|.
In addition, assume that
µ1(m̄1 + m̄2) + µ2(n̄1 + n̄2) < 1,
where µ1 and µ2 are given by (3.5) and (3.6) respectively. Then the boundary-value problem
(1.1) – (1.3) has a unique solution.
ISSN 1562-3076. Нелiнiйнi коливання, 2016, т . 19, N◦ 3
308 B. AHMAD, S. K. NTOUYAS
Example 3.1. Consider the following system of coupled fractional differential equations
with uncoupled slit-strips integral boundary conditions
cD5/4x(t) =
|x(t)|
24(1 + |x(t)|)
+
1
20
tan−1 y + 1, t ∈ [0, 1],
cD3/2y(t) =
1
35
sinx(t) +
1
25
y(t) + 4, t ∈ [0, 1],
(3.7)
x(0) = 0, x(1/2) =
1/3∫
0
x(s)ds+
1∫
2/3
x(s)ds,
y(0) = 0, y(1/2) =
1/3∫
0
y(s)ds+
1∫
2/3
y(s)ds.
Here q = 5/4, p = 3/2, a = 1, b = 1, ζ = 1/2, η = 1/3, ξ = 2/3. With the given values, it
is found that A = 1/6, m̄1 = 1/24, m̄2 = 1/20, n̄1 = 1/35, n̄2 = 1/25, µ1 ' 4.716276, µ2 '
' 3.614087. In consequence, µ1(m̄1+m̄2)+µ2(n̄1+n̄2) ' 0.680148 < 1.Thus all the conditions
of Theorem 3.1 are satisfied. Therefore, there exists a unique solution for the problem (3.7)
on [0, 1].
The second result dealing with the existence of solutions for the problem (1.1) – (1.3) is
analogous to Theorem 2.2 and is given below.
Theorem 3.2. Assume that there exist real constants ρi, νi ≥ 0, i = 1, 2, and ρ0 > 0, ν0 > 0
such that for any xi ∈ R, i = 1, 2, we have
|f(t, x1, x2)| ≤ ρ0 + ρ1|x1|+ ρ2|x2|,
|g(t, x1, x2)| ≤ ν0 + ν1|x1|+ ν2|x2|.
In addition it is assumed that
µ1ρ1 + µ2ν1 < 1 and µ1ρ2 + µ2ν2 < 1,
where µ1 and µ2 are given by (3.5) and (3.6) respectively. Then the boundary-value problem
(1.1) – (1.3) has at least one solution.
Proof. Setting
µ0 = min{1− (µ1ρ1 + µ2ν1), 1− (µ1ρ2 + µ2ν2)}, ρi, νi ≥ 0, i = 1, 2,
with µ1 and µ2 given by (3.5) and (3.6) respectively, the proof is similar to that of Theorem 2.2.
So we omit it.
References
1. Podlubny I. Fractional differential equations. — San Diego: Acad. Press, 1999.
ISSN 1562-3076. Нелiнiйнi коливання, 2016, т . 19, N◦ 3
A COUPLED SYSTEM OF NONLOCAL FRACTIONAL DIFFERENTIAL EQUATIONS . . . 309
2. Kilbas A. A., Srivastava H. M., Trujillo J. J. Theory and applications of fractional differential equations //
North-Holland Math. Stud. — Amsterdam: Elsevier Sci. B. V., 2006. — 204.
3. Sabatier J., Agrawal O. P., Machado J. A. T. (Eds.) Advances in fractional calculus: theoretical developments
and applications in physics and engineering. — Dordrecht: Springer, 2007.
4. Baleanu D., Mustafa O. G. On the global existence of solutions to a class of fractional differential equations
// Comput. Math. Appl. — 2010. — 59. — P. 1835 – 1841.
5. Ahmad B., Nieto J. J. Riemann – Liouville fractional integro-differential equations with fractional nonlocal
integral boundary conditions // Boundary Value Problems. — 2011. — 2011, № 36. — 9 p.
6. Alsaedi A., Ntouyas S. K., Ahmad B. Existence results for Langevin fractional differential inclusions invol-
ving two fractional orders with four-point multiterm fractional integral boundary conditions // Abstrs Appl.
Anal. — 2013. — Art. ID 869837. — 17 p.
7. Baleanu D., Mustafa O. G., Agarwal R. P. On Lp-solutions for a class of sequential fractional differential
equations // Appl. Math. Comput. — 2011. — 218. — P. 2074 – 2081.
8. Graef J. R., Kong L., Kong Q. Application of the mixed monotone operator method to fractional boundary
value problems // Fract. Calc. Different. Calc. — 2011. — 2. — P. 554 – 567.
9. Akyildiz F. T., Bellout H., Vajravelu K., Van Gorder R. A. Existence results for third order nonlinear boundary
value problems arising in nano boundary layer fluid flows over stretching surfaces // Nonlinear Anal. Real
World Appl. — 2011. — 12. — P. 2919 – 2930.
10. Bai Z. B., Sun W. Existence and multiplicity of positive solutions for singular fractional boundary value
problems // Comput. Math. Appl. — 2012. — 63. — P. 1369 – 1381.
11. Sakthivel R., Mahmudov N. I., Nieto J. J. Controllability for a class of fractional-order neutral evolution
control systems // Appl. Math. and Comput. — 2012. — 218. — P. 10334 – 10340.
12. Agarwal R.P., O’Regan D., Stanek S. Positive solutions for mixed problems of singular fractional differential
equations // Math. Nachr. — 2012. — 285. — P. 27 – 41.
13. Wang J. R., Zhou Y., Medved M. Qualitative analysis for nonlinear fractional differential equations via
topological degree method // Topol. Methods Nonlinear Anal. — 2012. — 40. — P. 245 – 271.
14. Cabada A., Wang G. Positive solutions of nonlinear fractional differential equations with integral boundary
value conditions // J. Math. Anal. and Appl. — 2012. — 389. — P. 403 – 411.
15. Wang G., Ahmad B., Zhang L., Agarwal R. P. Nonlinear fractional integro-differential equations on unboun-
ded domains in a Banach space // J. Comput. and Appl. Math. — 2013. — 249. — P. 51 – 56.
16. Ahmad B., Ntouyas S. K., Alsaedi A. A study of nonlinear fractional differential equations of arbitrary
order with Riemann – Liouville type multistrip boundary conditions // Math. Probl. Eng. — 2013. — Art. ID
320415. — 9 p.
17. O’Regan D., Stanek S. Fractional boundary value problems with singularities in space variables // Nonlinear
Dynam. — 2013. — 71. — P. 641 – 652.
18. Liu Z., Lu L., Szántó I. Existence of solutions for fractional impulsive differential equations with p-Laplacian
operator // Acta Math. hung. — 2013. — 141. — P. 203 – 219.
19. Wang J. R., Zhou Y., Feckan M. On the nonlocal Cauchy problem for semilinear fractional order evolution
equations // Cent. Eur. J. Math. — 2014. — 12. — P. 911 – 922.
20. Graef J. R., Kong L., Wang M. Existence and uniqueness of solutions for a fractional boundary value problem
on a graph // Fract. Calc. Appl. Anal. — 2014. — 17. — P. 499 – 510.
21. Wang G., Liu S., Zhang L. Eigenvalue problem for nonlinear fractional differential equations with integral
boundary conditions // Abstrs Appl. Anal. — 2014. — Art. ID 916260. — 6 p.
22. Punzo F., Terrone G. On the Cauchy problem for a general fractional porous medium equation with variable
density // Nonlinear Anal. — 2014. — 98. — P. 27 – 47.
23. Liu X., Liu Z., Fu X. Relaxation in nonconvex optimal control problems described by fractional differential
equations // J. Math. Anal. and Appl. — 2014. — 409. — P. 446 – 458.
24. Zhai C., Xu L. Properties of positive solutions to a class of four-point boundary value problem of Caputo
fractional differential equations with a parameter // Commun. Nonlinear Sci. Numer. Simul. — 2014. — 19. —
P. 2820 – 2827.
ISSN 1562-3076. Нелiнiйнi коливання, 2016, т . 19, N◦ 3
310 B. AHMAD, S. K. NTOUYAS
25. Ahmad B., Agarwal R. P. Some new versions of fractional boundary value problems with slit-strips conditions
// Boundary Value Problem. — 2014. — 2014, № 175. — 12 p.
26. Ahmad B., Nieto J. J. Existence results for a coupled system of nonlinear fractional differential equations
with three-point boundary conditions // Comput. Math. Appl. — 2009. — 58. — P. 1838 – 1843.
27. Su X. Boundary value problem for a coupled system of nonlinear fractional differential equations // Appl.
Math. Lett. — 2009. — 22. — P. 64 – 69.
28. Wang J., Xiang H., Liu Z. Positive solution to nonzero boundary values problem for a coupled system of
nonlinear fractional differential equations // Int. J. Different. Equat. — 2010. — 2010. — Article ID 186928. —
12 p.
29. Sun J., Liu Y., Liu G. Existence of solutions for fractional differential systems with antiperiodic boundary
conditions // Comput. Math. Appl. — 2012 /doi: 10.1016/j.camwa.2011.12.083
30. Ntouyas S. K., Obaid M. A coupled system of fractional differential equations with nonlocal integral boundary
conditions // Adv. Different. Equat. — 2012. — 130.
31. Faieghi M., Kuntanapreeda S., Delavari H., Baleanu D. LMI-based stabilization of a class of fractional-order
chaotic systems // Nonlinear Dynam. — 2013. — 72. — P. 301 – 309.
32. Senol B., Yeroglu C. Frequency boundary of fractional order systems with nonlinear uncertainties // J.
Franklin Inst. — 2013. — 350. — P. 1908 – 1925.
33. Otsuki T. Diffraction by multiple slits // JOSA A. — 1990. — 7. — P. 646 – 652.
34. Asghar S., Hayat T., Ahmad B. Acoustic diffraction from a slit in an infinite absorbing sheet // Jap. J. Indust.
Appl. Math. — 1996. — 13. — P. 519 – 532.
35. Mellow T., Karkkainen L. On the sound fields of infinitely long strips // J. Acoust. Soc. Amer. — 2011. —
130. — P. 153 – 167.
36. Lee D. J., Lee S. J., Lee W. S., Yu J. W. Diffraction by dielectric-loaded multiple slits in a conducting plane:
TM case // Prog. Electromagnet. Res. — 2012. — 131. — P. 409 – 424.
37. Granas A., Dugundji J. Fixed point theory. — New York: Springer-Verlag, 2003.
Received 14.01.15
ISSN 1562-3076. Нелiнiйнi коливання, 2016, т . 19, N◦ 3
|